scholarly journals Simple and subdirectly irreducibles bounded distributive lattices with unary operators

2006 ◽  
Vol 2006 ◽  
pp. 1-20 ◽  
Author(s):  
Sergio Arturo Celani
Keyword(s):  
Distributive Lattices ◽  
Heyting Algebras ◽  
Subdirectly Irreducible ◽  
Modal Algebras ◽  
Heyting Algebras With Operators ◽  
Subdirectly Irreducibles ◽  
Subdirectly Irreducible Algebras

We characterize the simple and subdirectly irreducible distributive algebras in some varieties of distributive lattices with unary operators, including topological and monadic positive modal algebras. Finally, for some varieties of Heyting algebras with operators we apply these results to determine the simple and subdirectly irreducible algebras.

scholarly journals On some small varieties of distributive Ockham algebras

1984 ◽  
Vol 25 (2) ◽  
pp. 175-181 ◽  
Author(s):  
R. Beazer
Keyword(s):  
Duality Theory ◽  
Basic Method ◽  
Distributive Lattices ◽  
Algebraic Proof ◽  
Subdirectly Irreducible ◽  
Topological Duality ◽  
Hasse Diagrams ◽  
Ockham Algebras ◽  
Subdirectly Irreducibles ◽  
Subdirectly Irreducible Algebras

J. Berman [2] initiated the study of a variety k of bounded distributive lattices endowed with a dual homomorphic operation paying particular attention to certain subvarieties km, n. Subsequently, A. Urquhart [8] named the algebras in k distributive Ockham algebras, and developed a duality theory, based on H. A. Priestley's order-topological duality for bounded distributive lattices [6], [7]. Amongst other things, Urquhart described the ordered spaces dual to the subdirectly irreducible algebras in Sif. This work was developed further still by M. S. Goldberg in his thesis and the paper [5]. Recently, T. S. Blyth and J. C. Varlet [3], in abstracting de Morgan and Stone algebras, studied a subvariety MS of the variety k1.1. The main result in [3]is that there are, up to isomorphism, nine subdirectly irreducible algebras in MS and their Hasse diagrams are exhibited. The methods employed in [3] are purely algebraic and can be generalized to show that, up to isomorphism, there are twenty subdirectly irreducible algebras in k1.1. In section 3 of this paper, we take a short cut to this result by utilizing the results of Urquhart and Goldberg. Our basic method is simple: the results of Goldberg [5] are applied to k1,1 to produce a certain eight-element algebra B1 in k1,1, whose lattice reduct is Boolean and whose subalgebras are, up to isomorphism, precisely the subdirectly irreducibles in k1.1. We then pick out of the list of twenty such algebras those belonging to the variety MS. In section 4, we sketch a purely algebraic proof along the lines followed by Blyth and Varlet in [3].

scholarly journals Distributive Ockham algebras: free algebras and injectivity

1981 ◽  
Vol 24 (2) ◽  
pp. 161-203 ◽  
Author(s):  
Moshe S. Goldberg
Keyword(s):  
Distributive Lattices ◽  
Free Algebras ◽  
Subdirectly Irreducible ◽  
Irreducible Algebra ◽  
Subdirectly Irreducible Algebra ◽  
Ockham Algebras ◽  
Subdirectly Irreducibles ◽  
Priestley's Duality

This paper centres around the variety 0 of distributive Ockham algebras, and those subvarieties of 0 which are generated by a single finite subdirectly irreducible algebra A. We use H.A. Priestley's duality for bounded distributive lattices throughout. First, intrinsic descriptions of the duals of certain finite subdirectly irreducibles are given; these are later used to determine projectives in the dual categories. Next, left adjoints to the forgetful functors from 0 and Var(A) into bounded distributive lattices are obtained, thereby allowing us to describe all free algebras and coproducts of arbitrary algebras. Finally, by applying the duality, we characterize injectivity in Var(A) for each finite subdirectly irreducible algebra A.

FAST ISOMORPHISM TESTING IN ARITHMETICAL VARIETIES

2003 ◽  
Vol 13 (04) ◽  
pp. 499-506
Author(s):  
TOMASZ A. GORAZD
Keyword(s):  
Finite Fields ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Boolean Algebras ◽  
Heyting Algebras ◽  
Subdirectly Irreducible ◽  
Finite Algebras ◽  
Element Chain ◽  
Subdirectly Irreducible Algebras

Let [Formula: see text] be a finitely generated, arithmetical variety such that all subdirectly irreducible algebras from [Formula: see text] have linearly ordered congruences. We show that there is a polynomial time algorithm that tests the existing of an isomorphism between any two finite algebras from [Formula: see text]. This includes the following classical structures in algebra: • Boolean algebras. • Varieties of rings generated by finitely many finite fields. • Varieties of Heyting algebras generated by an n–element chain.

scholarly journals Bitopological duality for distributive lattices and Heyting algebras

2010 ◽  
Vol 20 (3) ◽  
pp. 359-393 ◽  
Author(s):  
GURAM BEZHANISHVILI ◽  
NICK BEZHANISHVILI ◽  
DAVID GABELAIA ◽  
ALEXANDER KURZ
Keyword(s):  
Boolean Algebras ◽  
Distributive Lattices ◽  
Heyting Algebras ◽  
Priestley Spaces ◽  
Spectral Spaces ◽  
Algebraic Concepts ◽  
Stone Spaces

We introduce pairwise Stone spaces as a bitopological generalisation of Stone spaces – the duals of Boolean algebras – and show that they are exactly the bitopological duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many algebraic concepts important in the study of distributive lattices. We also give new bitopological and spectral dualities for Heyting algebras, thereby providing two new alternatives to Esakia's duality.

Frontal operators in distributive lattices with a generalized implication

2015 ◽  
Vol 08 (03) ◽  
pp. 1550039
Author(s):  
Sergio A. Celani ◽  
Hernán J. San Martín
Keyword(s):  
Distributive Lattice ◽  
Unary Operation ◽  
Duke Math ◽  
Distributive Lattices ◽  
Heyting Algebras ◽  
Bounded Distributive Lattice ◽  
Modal Lattices ◽  
Weak Heyting Algebras ◽  
San Martín

We introduce a family of extensions of bounded distributive lattices. These extensions are obtained by adding two operations: an internal unary operation, and a function (called generalized implication) that maps pair of elements to ideals of the lattice. A bounded distributive lattice with a generalized implication is called gi-lattice in [J. E. Castro and S. A. Celani, Quasi-modal lattices, Order 21 (2004) 107–129]. The main goal of this paper is to introduce and study the category of frontal gi-lattices (and some subcategories of it). This category can be seen as a generalization of the category of frontal weak Heyting algebras (see [S. A. Celani and H. J. San Martín, Frontal operators in weak Heyting algebras, Studia Logica 100(1–2) (2012) 91–114]). In particular, we study the case of frontal gi-lattices where the generalized implication is defined as the annihilator (see [B. A. Davey, Some annihilator conditions on distributive lattices, Algebra Universalis 4(1) (1974) 316–322; M. Mandelker, Relative annihilators in lattices, Duke Math. J. 37 (1970) 377–386]). We give a Priestley’s style duality for each one of the new classes of structures considered.

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